When I say "Apparently, xyz", does that imply one of the following, and if so, which one?

From observation, I believe xyz to be true, but I leave open the possibility that I might be wrong.

I believe xyz is not true, but I am aware that it may seem so, and want to point that out.

Merriam-Webster gives "open to view", "clear or manifest to the understanding" and "appearing as actual to the eye or mind", which to me seems to lean towards the first interpretation, but when discussing synonyms it also states

It can be used sarcastically, in which case the literal statement is in contravention of fact. Example: "Apparently no one is going to offer me a drink tonight." [Said petulantly, waiting to be offered a drink.] Absent that, though, no negativity attaches to the word unless further explanation is offered.
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RobustoAug 31 '12 at 21:31

1

... "apparently" is apparently unfathomable to some people! So I suggest you to not use it ...
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Elberich SchneiderAug 31 '12 at 21:40

@Robusto: There can be irony without the sarcasm too. The Fluxion's writers tend to be a little over-optimistic. Apparently, I died last night.Apparently (at least here) is being used as a modal pragmatic marker, referencing or at least warning of the source of the assertion (/ allegation) and hence fulfilling a hedging function.
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Edwin AshworthJan 3 '13 at 12:35

In the US, most listeners would probably take away the first interpretation.

The term apparently is a hedge to indicate that the speaker is not certain or does not have irrefutable evidence of the position, but either believes it is so or is willing to accept it for the sake of argument. It is akin to saying

It [now] appears to me that . . .

If one did not believe something was so, but wished to consider or discuss it, she or he might say Allegedly, xyz . . which implies that someone else has averred it, but the speaker is, or may be, doubtful. While reporters (and lawyers) use this locution to appear neutral, most others using it would likely be indicating skepticism. It is more like saying

Someone else is claiming that . . . [and I am doubtful, or at best neutral]

I sometimes use "apparently" in biomedical articles when there is statistically significant (P < 0.05) but inconclusive evidence that X causes Y. It's a substitute for X seems to cause Y or X may cause Y (X apparently causes Y).

Yes, another allowable usage. Here the modal pragmatic marker is being used to comment on the researcher's confidence in the validity of a hypothesis, rather than the reliability of a source of information.
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Edwin AshworthJan 3 '13 at 12:59